fluid_mechanics

272 Chapter 6 ■ Differential Analysis of Fluid Flow
y
^
e
θP ^e
v r
θ
r
v z
P
v r
^e
z
x
θ
z
F I G U R E 6.6 The representation of
velocity components in cylindrical polar coordinates.
For some problems,
velocity components
expressed in cylindrical
polar coordinates
will be
convenient.
6.2.2 Cylindrical Polar Coordinates
For some problems it is more convenient to express the various differential relationships in cylindrical
polar coordinates rather than Cartesian coordinates. As is shown in Fig. 6.6, with cylindrical coordinates
a point is located by specifying the coordinates r, u, and z. The coordinate r is the radial distance from
the z axis, u is the angle measured from a line parallel to the x axis 1with counterclockwise taken as
positive2, and z is the coordinate along the z axis. The velocity components, as sketched in Fig. 6.6,
are the radial velocity, v r , the tangential velocity, v u , and the axial velocity, Thus, the velocity at
some arbitrary point P can be expressed as
V v r ê r v u ê u v z ê z
v z .
(6.32)
where ê r , ê u , and ê z are the unit vectors in the r, u, and z directions, respectively, as are illustrated
in Fig. 6.6. The use of cylindrical coordinates is particularly convenient when the boundaries of
the flow system are cylindrical. Several examples illustrating the use of cylindrical coordinates will
be given in succeeding sections in this chapter.
The differential form of the continuity equation in cylindrical coordinates is
0r
0t 1 01rrv r 2
1 01rv u 2
01rv z2
0
r 0r r 0u 0z
(6.33)
This equation can be derived by following the same procedure used in the preceding section 1see
Problem 6.202. For steady, compressible flow
1 01rrv r 2
1 01rv u 2
01rv z2
0
r 0r r 0u 0z
(6.34)
For incompressible fluids 1for steady or unsteady flow2
1 01rv r 2
1 0v u
r 0r r 0u 0v z
0z 0
(6.35)
6.2.3 The Stream Function
Steady, incompressible, plane, two-dimensional flow represents one of the simplest types of flow
of practical importance. By plane, two-dimensional flow we mean that there are only two velocity
components, such as u and v, when the flow is considered to be in the x–y plane. For this flow
the continuity equation, Eq. 6.31, reduces to
0u
0x 0v
0y 0
(6.36)